TITLE: What is an element of an iterated tangent bundle?
QUESTION [6 upvotes]: An element of the tangent bundle $T M$ of a manifold is called a "(tangent) vector".  An element of its dual $T^* M$ is called a "covector" or a "1-form".  An element of the exterior square $\Lambda^2(T M)$ is called a "bivector", and an element of $\Lambda^2(T^*M)$ is called a "2-form".  More generally, elements of the various tensor powers of $T M$ and $T^*M$ are called "tensors".
Is there a name for an element of an iterated tangent bundle $T^k M = T(T(\cdots (T M)))$?

REPLY [4 votes]: I called them "expansion" because they act on functions on $M$ by (Taylor) expanding them.
This is a generalization of derivation. 
See 37.6 of 

Ivan Kolár, Jan Slovák, Peter W. Michor: Natural operations in differential geometry. Springer-Verlag, Berlin, Heidelberg, New York, (1993) pdf.

Perhaps you need to skim the whole chapter VIII.